{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "8c25e491-fcbc-4b1a-a2a8-3b14fdb48981",
   "metadata": {
    "setTimeout": 120
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "from sympy import init_session\n",
    "init_session(quiet=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6eb18ea5-c956-4440-84da-c9ef2fb4d6a0",
   "metadata": {},
   "source": [
    "# Limits \n",
    "\n",
    "## Syllabus Goals\n",
    "Working on this through week ending October 9, 2022.  \n",
    "\n",
    "> _\"The problems are the solution.\"_ (Johnsin University B.S. Program Guide)\n",
    "\n",
    "Assignments:\n",
    "    \n",
    "* [Paul's Math Notes Limits Index](https://tutorial.math.lamar.edu/Classes/CalcI/LimitsIntro.aspx).\n",
    "* Paul's Math Notes on L'Hospital's Rule.\n",
    "* Review Asympotes vs Holes Professor Leonard Notes. Organize some sample problems around this.\n",
    "    \n",
    "Try for example on $$ \\frac{\\sin{x}}{x} $$ or other hard-to-do problems.  Inductively work with L'Hospital's rule.\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ead2227-b172-4947-a1e1-f357927919ad",
   "metadata": {},
   "source": [
    "## Different Types of Limits:\n",
    "\n",
    "* Non-rational polynomials are easy, can always do by substitution.  e.g. $\\lim_{x\\to2} 3x^2 + 2x - 9 = 7$.  Paul's math notes expresses this formally thus:  If p(a) is a polynomial, then $\\lim_{x\\to a} p(x) = p(a)$.\n",
    "\n",
    "### For rational polynomials:\n",
    "* If substituting gives non-zero in numerator and denominator, easy peasy, $\\lim_{x\\to2}\\frac{3x^2 + 2x - 9 = 24}{2x +3} = \\frac{7}{7} = 1$\n",
    "* Zero in numerator only is fine. So for example $\\lim_{x\\to2}\\frac{x^2 - 4}{3x + 7} = \\frac{0}{7} = 0$ Note also in Python, print(0/anything) = 0.0.\n",
    "* __Zero in denominator only is an asymptote or DNE because asymptotes are one-sided.__ That is, answer will be always be either $\\infty$, $- \\infty$, or $\\pm\\infty$ depending on the direction, right-hand limit, left-hand limit, etc. The sign is what tells you what's happening at each direction.  The proof is, if you're approaching zero, you're dividing by, for example $\\frac{1}{10},\\frac{1}{100},...$ Dividing by fraction = multiplying by reciprocal of denominator, ie. by $10, 100, 1000, 10000...$.  So you're going toward infinity. For example $\\lim_{x\\to2}\\frac{3x + 7}{x^2 - 4} = \\:?$.  See the [graph](https://www.desmos.com/calculator/zuea89ausp).  Because of this, __you have to check denominator by plugging in values > and < x (informal \"sign analysis\").__  Any easy values above and below are OK.  For example, given denom of $x^2 - 4$, if x is 1, get $1^2 - 4 = 1 - 4 = -3$, negative.  But for x = 3, get $3^2 - 4 = 9 - 4 = 5$, positive.  And sure enough, graph shows $\\lim_{x\\to0^{+}} = +\\infty $ and $\\lim_{x\\to0^{-}}= -\\infty$ but _the_ limit does not exist.\n",
    "\n",
    "(See limit notes below on functions at infinity -- relearned today in Goodman's Barron's Calculus section 3.3.)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "919b859c-8604-4a66-a057-349e9e7f4f9f",
   "metadata": {},
   "source": [
    "Continuity and Limits from Paul's Math Notes:\n",
    "## Definition of continuity\n",
    "A function $f(x)$ is continuous at $x = a$ if \n",
    "\n",
    "$$\n",
    "\\lim_{x\\to a} f(x) = f(a)\n",
    "$$\n",
    "\n",
    "A function is said to be continuous on the interval $[a,b]$ if it is continuous at each point on the interval. \n",
    "\n",
    "The corrolary of this is that if $f(x)$ is continuous at $x = a$, then the following is true:\n",
    "\n",
    "$$\n",
    "\\lim_{x\\to a} f(x) = \\lim_{x\\to a+} f(x) = \\lim_{x\\to a-} f(x) = f(a)\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "19526072-ec6c-4f76-be6b-b3af14903191",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7\n",
      "1\n",
      "0\n"
     ]
    }
   ],
   "source": [
    "# Checking math on above in SymPy\n",
    "from sympy.plotting import plot\n",
    "print(limit(3*x**2 + 2*x - 9, x, 2))\n",
    "print(limit((3*x**2 + 2*x - 9)/(2*x + 3), x, 2))\n",
    "print(limit((x**2 - 4)/(3*x + 7), x, 2))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24a52028-efe4-41bf-bbf5-3b7ca22e419e",
   "metadata": {},
   "source": [
    "## Weekly review\n",
    "\n",
    "* Sat did about 1 hour on above notes.\n",
    "* Sun 1.5 hours \n",
    "* Mon 3 hours.  Hard, and not much writing done.\n",
    "* Tues 2.5 Hours.\n",
    "* Wed 2.0 hours 1.5 reviewing limits and a Set Theory Youtube introduction\n",
    "* Thur 2-ish (began working again on Derivatives)\n",
    "* Fri TBD\n",
    "\n",
    "___ \n",
    "12 hours (Not counting Friday -- TBD)\n",
    "\n",
    "\n",
    "Getting to three hours per day didn't really work, and cut a whole in writing (see Monday).  Mostly the limits work went well, lots of problems done early in the week. Slowed down on quadratic review.  Did not really do much with L'Hopital yet."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "453866c1-ccf5-485c-b745-e140c13da3fc",
   "metadata": {},
   "source": [
    "## Limits Notes (Original)\n",
    "\n",
    "This part of notes from [Calc 1 3.5 Limits of Functions at Infinity](https://www.youtube.com/watch?v=-PYebK8DKPc&t=1s). When numerator equals zero, that's a point, zero, fine.  When denominator equals zero, that's a discontinuity. \n",
    "\n",
    "It's a removeable discontinuity if you can factor it; you can tell this often if you get $\\frac{0}{0}$. Otherwise it's an asymptote -- can't be canceled, only in denom.\n",
    "\n",
    "We should be able to show this in Sympy."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fba1d05f-a054-4c8d-bca1-d31e92ce9256",
   "metadata": {},
   "source": [
    "$$\n",
    "y = mx +b\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "24deca57-6668-4d44-a072-6140e437394f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{x}{\\left(x - 1\\right) \\left(x + 3\\right)}$"
      ],
      "text/plain": [
       "       x       \n",
       "───────────────\n",
       "(x - 1)⋅(x + 3)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = x / ( (x + 3) * (x - 1) )\n",
    "f"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a60305ee-d4e9-41e3-a606-33bc254f25ec",
   "metadata": {},
   "source": [
    "This should have two asymptotes at + 1 and -3 (NOT at zero, numerator is fine)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "9eb9e96e-3040-45bf-a7da-a2015c006f6a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x10fb0ded0>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "plot(f, xlim=(-4, 2), ylim=(-10, 10))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "061f5d56-8dae-4afe-b4d9-253c053be1b1",
   "metadata": {},
   "source": [
    "For the following function, the x-1 gets cancelled out.  So this has ONE asymptote at -3.s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "ea52284e-4915-40a3-851d-ae1ec6bd0b36",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAAtCAYAAADoSujCAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACoklEQVRoBe2Z7XETMRBAbcYFJNCBS0hMB6YDAhWQdEAmv+x/TOiAUALpIO4gJCVQgnEH5j0h3Rw2Fzs+S0lmbmfWkna12k/p7uT+crns5YLpdHrE2j/AY/qLHHoG+14UQw9Y8zs4B0fgEMwGORww0idajDOfacxCNniVbeVCC3cOFAp0o5ouA42hKcToMlAo0I1qugw0hqYQI3cG3kQ/Xufyp5/jZY5XCF/ghDHou9E9+Au8gXdFuzfI4sDerNtiodwltIUJ7aZ0DrSLX3vpLgPtY9huhS4D7eLXXvrFZ6A/mUzy3au0D/DGFbon8cYQZZ7w4vdA50DmCtm4/N6vFjdqfOQEvh+GiJxFMb8tHF9Cn0l71qcQRmqwxiYHevTfQ/OD6YT+dfY9gJIxeIrCXUC502h0kg+RZ3AhIbsD6DCK4i7gp+giYpDHGccVPOs9gLFG+7Cylk4tG9+kVw7AMEop1W/pfwLdMB9B4ZY513+7T/OLfi8JLsEz+uFyoHJABsSwWWj9Y8J/WTT6HNRbBZ/EAfT7J4nGG1jL6icYIDjABCOvgQmsM3e7WRC816nzA7HUD/ZptNijr113tOEUCscogyMwTIiTjPgI2rHjbYC5yhilVUiXWvNVBuN75MLfUf/hNZKQ+Q3Tkj9MGaiMj1IfaL/E/lYNi1ZndV0AuhEb0n6t07fpI2Pp9GhX7bOEDNZ47Rhlsgy9q+od2oEIrTTcodByadQ9iEyfbOnxbCQX0L0KTHDB+DwNCrYLdM3QbVuHURzMzEBIBe08OvNPrUKTfxsFSjcG7aauFHssSTPiUboY0PFh4ZmqoT2IMryEdVOaQh2rysk5pQC9V+A42pLU+mx6B63MyxyKdt7EyeKH2rVN/NDkHXnW72oN77jUutgfZzfScgJiTvoAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle \\frac{1}{x + 3}$"
      ],
      "text/plain": [
       "  1  \n",
       "─────\n",
       "x + 3"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "g = (x - 1) / ((x + 3) * (x - 1) )\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "ac6f479c-4bba-4b72-a64c-bb795b114e2d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x10f1b0d90>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(g, xlim=(-4, 2), ylim=(-10, 10))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cf78e875-0389-4838-b699-1bd1124aa343",
   "metadata": {},
   "source": [
    "To determine which way it's going, just plug in numbers along interval.  So given asymptotes at -3, 1, can plug in values for intervals > 1 and < -3.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "09e670e7-5fc8-4c8f-bce5-7e3339d9d9e2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ - \\frac{4}{5}, \\  0.133333333333333, \\  -0.285714285714286, \\  \\frac{2}{5}\\right]$"
      ],
      "text/plain": [
       "[-4/5, 0.133333333333333, -0.285714285714286, 2/5]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[f.subs(x, i) for i  in (-4, -.5, .5, 2)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "18415ae9-9e90-4616-b555-369bb348761f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle - \\frac{4}{5}$"
      ],
      "text/plain": [
       "-4/5"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.subs(x, -4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "d1c250c6-2372-473a-b4d1-ff327504a448",
   "metadata": {},
   "outputs": [
    {
     "ename": "SyntaxError",
     "evalue": "invalid syntax (60756867.py, line 1)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;36m  Cell \u001b[0;32mIn[9], line 1\u001b[0;36m\u001b[0m\n\u001b[0;31m    So limit doesn't exist at -3 or 1, because not going in same direction.  Limit DOES exist at zero, because it's a point.\u001b[0m\n\u001b[0m       ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n"
     ]
    }
   ],
   "source": [
    "So limit doesn't exist at -3 or 1, because not going in same direction.  Limit DOES exist at zero, because it's a point."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "88950e3a-09b4-4681-bd4f-621c416d6108",
   "metadata": {},
   "source": [
    "## Example\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d8e2e851-6187-4a89-8b64-f5497b148c05",
   "metadata": {},
   "outputs": [],
   "source": [
    "h = t**3 / (t**2 -1)**2\n",
    "h"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e202daa-8ff9-42bc-b158-ea942aae3c7c",
   "metadata": {},
   "source": [
    "By factoring t squared - 1, get:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d2d4dfde-5818-4aae-af1f-ca54a6ea555c",
   "metadata": {},
   "outputs": [],
   "source": [
    "(t + 1)*(t- 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62c18761-461d-418f-b436-b1c31b82c5a3",
   "metadata": {},
   "source": [
    "So discontinuities at 1, -1. these are asymptotes.  So we could substitute:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "885483ba-a80e-4907-80e8-8a2051e6f3b8",
   "metadata": {},
   "outputs": [],
   "source": [
    "[h.subs(t, i) for i  in (-2, -.5, .5, 2)]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4cae9387-cbe6-4860-8376-b40b804768a1",
   "metadata": {},
   "source": [
    "So my guess is we'll get limits at negative infinity, infinity.\n",
    "I was right about guess I think but graph looks like hell in this case.  Looks better on [Desmos](https://www.desmos.com/calculator/qwyl3yehs1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e893a6bf-8f14-43f6-b36a-14641c6af202",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "plot(h, xlim=(-3, 3), ylim=(-100, +100), )\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "283a3041-0e1b-4400-9f43-744b76dd2282",
   "metadata": {},
   "source": [
    "Next he asks what happens as x --> + oo or x --> - oo, for example:\n",
    "\n",
    "$$\n",
    "\\lim_{x \\to -\\infty} \\frac{1}{x}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\lim_{x \\to \\infty} \\frac{1}{x}\n",
    "$$\n",
    "\n",
    "Using a table x, f(x) for 1, 10, 100, 10000000, showed that limit approaching positive infinity is zero.  Same for negative infinity case.\n",
    "\n",
    "So if f(x) gets \"really close\" to a certain number as x -> +/- oo, then the LIMIT EXISTS.  But what does it mean?  So you end up with a HORIZONTAL asymptote.\n",
    "\n",
    "So: \n",
    "\n",
    "* Vertical asymptote:  When you have a non-removal discontinuity.\n",
    "* Horizontal asymptote: if you have a limit for f(x) as x gets really close to a certain number as x -> +/- infinity.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69108bab-2d07-4d60-a664-d4a5b8e3be57",
   "metadata": {},
   "source": [
    "Also, because we can pull out any exponent, any constant / some variable to the nth power e.g. \n",
    "\n",
    "$$\n",
    "\\frac{1}{x^n}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0538ce78-0ec3-4ff1-a04e-d55f75fa1d49",
   "metadata": {},
   "source": [
    "The limit of polynomial as x -> +/- oo:\n",
    "\n",
    "Polynomials always have tails going up or down or both.  Never actually go to a number, they go to infinity in one direction or other as x goes to +/- infinity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c1ffe065-c912-4ab6-9798-d7492b92951e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Scratch\n",
    "cos(0)\n",
    "cos(.00001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b05ff760-21b6-4809-83ea-744d9e5f4b32",
   "metadata": {},
   "outputs": [],
   "source": [
    "cos(.001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eb7a36c2-da5e-4e94-b8f2-5bcfbd91342d",
   "metadata": {},
   "outputs": [],
   "source": [
    "cos(-.001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2d5d8b2b-e7f6-4b04-9693-957b1cb54327",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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